Chapter 8.2, Problem 8E

(a)

To determine

For each of part Exercise 7 decide whether $f\prec gorf\simeq g$. Use Propositions 8.2.7 and 8.2.8, and show that $f=O(g)$ for $f(n)=5n,g(n)=n^3$ and $f,g:N\to R$.

(b)

To determine

For each of part Exercise 7 decide whether $f\prec gorf\simeq g$. Use Propositions 8.2.7 and 8.2.8, and show that $f=O(g)$ for $f(n)=17n^4+8n^3+5n^2+6n+1,g(n)=n^4$ and $f,g:N\to R$.

(c)

To determine

For each of part Exercise 7 decide whether $f\prec gorf\simeq g$. Use Propositions 8.2.7 and 8.2.8, and show that $f=O(g)$ for $f(n)=8n^3+4n^2+5n+1,g(n)=3n^4+6n^2+8n+2$ and $f,g:N\to R$.

(d)

To determine

For each of part Exercise 7 decide whether $f\prec gorf\simeq g$. Use Propositions 8.2.7 and 8.2.8, and show that $f=O(g)$ for $f(n)=2^n,g(n)=3^n$ and $f,g:N\to R$.

(e)

To determine

For each of part Exercise 7 decide whether $f\prec gorf\simeq g$. Use Propositions 8.2.7 and 8.2.8, and show that $f=O(g)$ for $f(n)=2n^2-3n+1,g(n)=n^2+1$ and $f,g:N\to R$.

(f)

To determine

For each of part Exercise 7 decide whether $f\prec gorf\simeq g$. Use Propositions 8.2.7 and 8.2.8, and show that $f=O(g)$ for $f(n)=2n^2-3n+5,g(n)=n^2-7n$ and $f,g:N\to R$.

(g)

To determine

For each of part Exercise 7 decide whether $f\prec gorf\simeq g$. Use Propositions 8.2.7 and 8.2.8, and show that $f=O(g)$ for $f\left(\text{n}\right)={\text{log}}_{\text{2}}{n}^5,g\left(\text{n}\right)=n$ and $f,g:N\to R$.